Vietoris thickenings and complexes have isomorphic homotopy groups

Abstract

We study the relationship between metric thickenings and simplicial complexes associated to coverings of metric spaces. Let U be a cover of a separable metric space X by open sets with a uniform diameter bound. The Vietoris complex V(U) contains all simplices with vertex set contained in some U ∈ U , and the Vietoris metric thickening V m (U) is the space of probability measures with support in some U ∈ U , equipped with an optimal transport metric. We show that V m (U) and V(U) have isomorphic homotopy groups in all dimensions. In particular, by choosing the cover U appropriately, we get isomorphisms between the homotopy groups of Vietoris-Rips metric thickenings and simplicial complexes π k (VR m < (X; r)) ∼ = π k (VR<(X; r)), as well as between the homotopy groups of Čech metric thickenings and simplicial complexes π k ( Čm < (X; r)) ∼ = π k ( Č<(X; r)), for all integers k ≥ 0.

The homotopy category of acyclic complexes of pure-projective modules

“…The relationship between |VRpX; rq| and VR m pX; rq was studied in further detail in [4,5]. Adams, Frick, and Virk showed in [4] that |VRpX; rq| and VR m pX; rq have isomorphic homotopy groups for any separable metric space X and arbitrary r ą 0.…”

“…The relationship between |VRpX; rq| and VR m pX; rq was studied in further detail in [4,5]. Adams, Frick, and Virk showed in [4] that |VRpX; rq| and VR m pX; rq have isomorphic homotopy groups for any separable metric space X and arbitrary r ą 0. More precisely, it was shown that the Vietoris complex VpU q and Vietoris metric thickening V m pU q, which generalize VRpX; rq and VR m pX; rq respectively, have isomorphic homotopy groups whenever U is a uniformly bounded open cover of a separable metric space X.…”

“…Ultimately, one would like to know when |VpU q| and V m pU q have the same homotopy type, as this could allow the homotopy type of |VpU q| to be studied through metric techniques applied to V m pU q. As a step in this direction, our main result is to address Question 7.2 of [4] and show that the natural bijection |VpU q| Ñ V m pU q is a weak homotopy equivalence.…”

“…The authors of [4] asked whether |VpU q| and V m pU q have isomorphic homology groups. This was shown to be true in [13] through an argument involving the Mayer-Vietoris spectral sequence anticipated by the authors of [4].…”

“…The authors of [4] asked whether |VpU q| and V m pU q have isomorphic homology groups. This was shown to be true in [13] through an argument involving the Mayer-Vietoris spectral sequence anticipated by the authors of [4]. However this fact is also a direct consequence of Theorem 1, as a weak homotopy equivalence induces an isomorphism of (co)homology groups.…”